Characterization of anysotropic Liouville-type mixed spaces and its application

Within the framework of the method of hypersingular integrals we obtain a characterization of the anysotropic spaces $L_{\bar{p},\bar{r}}^{\bar{\alpha }}$. These spaces are defined to consist of functions $f(x)\in L_{\bar{r}}$ for which $F^{-1}({\sum }_{j=1}^n\mathrm{\mid }{\xi }_j{\mathrm{\mid }}^{{\alpha }_j})Ff\in L_{\bar{p}},{\alpha }_j>0.$ The mentioned characterization is applied to prove the denseness of the class $C_0^{\mathrm{\infty }}$ in $L_{\bar{p},\bar{r}}^{\bar{\alpha }}$.